Spin-Stratum Type-Rigidity: Orthogonality of the ADE Gate and Closure of the Chiral Lift

The generation-split value $\epsilon=\tfrac{1}{10}$ is often presented as a derived prediction of the spectral stratigraphy of the corpus. This note isolates its exact logical status. First, $\epsilon=\tfrac{1}{10}$ is a theorem modulo the ADE case-selection gate: once a case is selected whose class-complete spectrum carries the external level ratio $3:2$, the generation-deficit normal form forces $\epsilon=\tfrac{1}{10}$ and $\kappa=\tfrac{5}{12}$ algebraically, with no further input. Second, the first-principles status of $\epsilon$ is therefore exactly the first-principles status of the gate. We then characterise the obstruction to deriving the gate as an arithmetic orthogonality. The selection of the $\sqrt{5}$-locus that carries the $3:2$ ratio requires the nontrivial element of ${\mathrm{Gal}}(\mathbb{Q}(\sqrt{5})/\mathbb{Q})$, the action $\sqrt{5}\mapsto-\sqrt{5}$. The only field automorphism supplied by the foundations is the Born–Infeld parity $c\mapsto q-c$, which is the restriction of complex conjugation $\zeta_q\mapsto\zeta_q^{-1}$ and fixes $\sqrt{5}$. In the compositum $K_q=\mathbb{Q}(\zeta_q,\sqrt{5})$ the two actions lie in distinct direct factors of the Galois group, hence are arithmetically orthogonal. Consequently the gate cannot be discharged from the present axioms without an additional arithmetic principle at the spin stratum. A bounded character-table audit of the binary icosahedral group $2I\cong\mathrm{SL}(2,5)$ establishes the type-rigidity the gate needs: the operations available at the spin stratum of the present corpus construction all fix $\mathbb{Q}(\sqrt{5})$, so their generated group lies in the $\sqrt{5}$-fixing subgroup of ${\mathrm{Gal}}(K_q/\mathbb{Q})$, which by closure cannot manufacture the outer automorphism that realises $\sqrt{5}\mapsto-\sqrt{5}$. The no-go for the ADE gate is therefore unconditional in the present construction; the same rigidity closes the chiral orientation note, the Lorentzian chiral lift inducing no action on $\mathbb{Q}(\sqrt{5})$ and hence $[\mathrm{H\text{-}orient}]$ unconditionally within the present spin stratum. The abstract type-rigidity of the full recursive tower is then established under the corpus notion of an admissible fibre operation: recursive non-injectivity multiplies the vertical parity type already present and cannot manufacture the horizontal Galois transport, so it preserves the binary-icosahedral arithmetic type.